We have seen that functions are special types of relations. When dealing with functions we often use function notation to emphasise that we are dealing with a special case, to highlight the independent and dependent variable and for ease of notation with substitution and calculus.
When we are writing in function notation, instead of writing "$$y=", we write "$$f(x)=". This can be interpreted as "$$f is a function of the variable $$x" and read as "$$f of $$x". Common letters to use for general functions are lower case $$f, $$g and $$h. However, any letter can be used and we can use variables that have meaning in context.
Instead of $$y=2x+1 we could write $$f(x)=2x+1, here $$f is a function of the variable $$x which follows the rule double $$x and add $$1.
$$P(t)=200×0.8t, here population is a function of time.
$$H(d)=30−2d−30d2, here height is a function of distance.
These are all examples of just a different way to write '$$y= …' notation as a function. The letter in the bracket on the left is the input and the right hand side gives us a rule for the output.
Function notation also allows for shorthand for substitution. If $$y=3x+2 and we write this in function notation as $$f(x)=3x+2, then the question 'what is the value of $$y when $$x is $$5?' can be asked simply as 'what is $$f(5)?' or 'Evaluate $$f(5).'
If $$A(x)=x2+1 and $$Q(x)=x2+9x, find:
(a) $$A(5)
Think: This means we need to substitute $$5 in for $$x in the $$A(x) equation.
Do:
$$A(5) | $$= | $$52+1 |
$$= | $$26 |
(b) $$Q(6)
Think: This means we need to substitute $$6 in for $$x in the $$Q(x) equation.
Do:
$$Q(6) | $$= | $$62+9×6 |
$$= | $$36+54 | |
$$= | $$90 |
(c) $$A(p)
Think: This means we need to substitute $$p in for $$x in the $$A(x) equation.
Do:
$$A(p) | $$= | $$p2+1 |
Consider the equation $$x−4y=8.
Assume that $$y is a function of $$x.
Rewrite the equation using function notation $$f(x).
Find the value of $$f(12).
Consider the function $$f(x)=2x3+3x2−4.
Evaluate $$f(0).
Evaluate $$f(14).
If $$Z(y)=y2+12y+32, find $$y when $$Z(y)=−3.
Write both solutions on the same line separated by a comma.
Consider the function $$g(x)=ax3−3x+5.
Determine $$g(k).
Form an expression for $$g(−k).
Is $$g(k)=g(−k)?
Yes
No
Is $$g(k)=−g(−k)?
Yes
No